| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Optimal estimator construction |
| Difficulty | Standard +0.3 This is a standard S4 question on unbiased estimators and minimum variance. Part (a) is routine verification using linearity of expectation. Part (b) requires using the unbiasedness constraint to eliminate b, then substituting into the variance formula—straightforward algebra. Part (c) uses basic calculus (differentiation) to minimize. All techniques are standard for this module with no novel insights required, making it slightly easier than average. |
| Spec | 5.05b Unbiased estimates: of population mean and variance |
A random sample of three independent variables $X_1, X_2$ and $X_3$ is taken from a distribution with mean $\mu$ and variance $\sigma^2$.
\begin{enumerate}[label=(\alph*)]
\item Show that $\frac{1}{3}X_1 + \frac{1}{3}X_2 + \frac{1}{3}X_3$ is an unbiased estimator for $\mu$. [3]
\end{enumerate}
An unbiased estimator for $\mu$ is given by $\hat{\mu} = aX_1 + bX_2$ where $a$ and $b$ are constants.
\begin{enumerate}[label=(\alph*)]
\item[(b)] Show that Var($\hat{\mu}$) = $(2a^2 - 2a + 1)\sigma^2$. [6]
\item Hence determine the value of $a$ and the value of $b$ for which $\hat{\mu}$ has minimum variance. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S4 Q6 [14]}}